| L(s) = 1 | − 2-s + 2.33·3-s + 4-s − 4.05·5-s − 2.33·6-s + 0.675·7-s − 8-s + 2.45·9-s + 4.05·10-s + 2.73·11-s + 2.33·12-s − 2.65·13-s − 0.675·14-s − 9.46·15-s + 16-s + 7.65·17-s − 2.45·18-s + 1.55·19-s − 4.05·20-s + 1.57·21-s − 2.73·22-s − 2.37·23-s − 2.33·24-s + 11.4·25-s + 2.65·26-s − 1.27·27-s + 0.675·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.34·3-s + 0.5·4-s − 1.81·5-s − 0.953·6-s + 0.255·7-s − 0.353·8-s + 0.817·9-s + 1.28·10-s + 0.823·11-s + 0.674·12-s − 0.735·13-s − 0.180·14-s − 2.44·15-s + 0.250·16-s + 1.85·17-s − 0.578·18-s + 0.356·19-s − 0.906·20-s + 0.344·21-s − 0.582·22-s − 0.495·23-s − 0.476·24-s + 2.28·25-s + 0.519·26-s − 0.245·27-s + 0.127·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.484227855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.484227855\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 2.33T + 3T^{2} \) |
| 5 | \( 1 + 4.05T + 5T^{2} \) |
| 7 | \( 1 - 0.675T + 7T^{2} \) |
| 11 | \( 1 - 2.73T + 11T^{2} \) |
| 13 | \( 1 + 2.65T + 13T^{2} \) |
| 17 | \( 1 - 7.65T + 17T^{2} \) |
| 19 | \( 1 - 1.55T + 19T^{2} \) |
| 23 | \( 1 + 2.37T + 23T^{2} \) |
| 29 | \( 1 + 7.93T + 29T^{2} \) |
| 31 | \( 1 - 6.83T + 31T^{2} \) |
| 37 | \( 1 - 8.43T + 37T^{2} \) |
| 41 | \( 1 + 1.85T + 41T^{2} \) |
| 43 | \( 1 - 3.52T + 43T^{2} \) |
| 47 | \( 1 + 7.66T + 47T^{2} \) |
| 53 | \( 1 - 5.03T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 1.22T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 - 7.37T + 89T^{2} \) |
| 97 | \( 1 - 4.07T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.494557183258515497850668877716, −8.357854296441435711395415246564, −8.010743013762906551647113038470, −7.57264777458356232391908219996, −6.72429432182349101052868389047, −5.21010469559534164632294263086, −3.88594763526097641645054785249, −3.52214137496757973917624141528, −2.42918002489419508352056988562, −0.923834286621207513799150621804,
0.923834286621207513799150621804, 2.42918002489419508352056988562, 3.52214137496757973917624141528, 3.88594763526097641645054785249, 5.21010469559534164632294263086, 6.72429432182349101052868389047, 7.57264777458356232391908219996, 8.010743013762906551647113038470, 8.357854296441435711395415246564, 9.494557183258515497850668877716
